I believe Web (Epiphany) is designed principally for the GNOME desktop environment and maybe this is a factor or do I need to design differently for these two browsers to make them display the fonts correctly in Linux?

I don't use either of those browsers myself, but in common with most browsers, somewhere in their settings they have a tick box for "Allow pages to choose their own fonts" which is usually selected by default. Before you start redesigning your website I would check to see if that option is still selected in your versions of Epiphany and Midori.

In Web (Epiphany) the default is set as 'Use system fonts' and the user options are not available, unticking this allows the user to define their own fonts.

In Midori, by default 'Always use my font choices' is unchecked, so you would expect that my default css styles would apply. However, there is also a field defined as 'Proportional Font Family' where you can change the font style and size. This field impacts on the web page without the need to tick the 'Always use my font choices' box? So this is probably overriding the css style.

Neither browser offers the option to configure an Arial font and I wonder if this part of the problem? A quick search on the web suggest that there is no Arial font in Linux.

Well in the great sea of browser users. I would think Midori & Epiphany are very very small group of fish to put too much time or worry into it. Midori is so new and evolving so expect major changes that may no longer work or not needed at all down the line.

Just a thought and hope you can resolve the issue quickly and easily..